1 / 15

CS 140 Lecture 6: Other Types of Gates

This lecture covers the concepts of universal gates, XOR gate, NAND/NOR gates, and block diagram transfers in combinational logic. It also discusses the properties and transformations of these gates.

campbellc
Télécharger la présentation

CS 140 Lecture 6: Other Types of Gates

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CS 140 Lecture 6: Other Types of Gates Professor CK Cheng

  2. Combinational Logic: Other Types of Gates • Universal Set of Gates • Other Types of Gates • XOR • NAND / NOR • Block Diagram Transfers

  3. Universal Set Universal Set: A set of gates such that every switching function can be implemented with gates in this set. Ex: {AND, OR, NOT} {AND, NOT} {OR, NOT}

  4. Universal Set Universal Set: A set of gates such that every Boolean function can be implemented with gates in this set. Ex: {AND, OR, NOT} {AND, NOT} OR can be implemented with AND & NOT gates a+b = (a’b’)’ {OR, NOT} AND can be implemented with OR & NOT gates ab = (a’+b’)’ {AND, OR} This is not universal.

  5. {NAND, NOR} {NOR} {XOR, AND} X 1 = X*1’ + X’*1 = X’ if constant “1” is available. 1

  6. Other Types of Gates 1) XOR X Y = XY’ + X’Y XOR X Y • Commutative X Y = Y X • Associative (X Y) Z = X (Y Z) • 1 X = X’ 0 X = 0X’ + 0’X = X • X X = 0, X X’ = 1

  7. e) if ab = 0 then a b = a + b Proof: If ab = 0 then a = a (b+b’) = ab+ab’ = ab’ b = b (a + a’) = ba + ba’ = a’b a+b = ab’ + a’b = a b f) X XY’ X’Y (X + Y) X = ?? To answer, we apply Shannon’s Expansion.

  8. Shannon’s Expansion (for switching functions) Formula: f (x,Y) = x * f (1, Y) + x’ * f (0, Y) Proof by enumeration: If x = 1, f (x,Y) = f (1, Y) : 1*f (1, Y) + 1’*f(0,Y) = f (1, Y) If x = 0, f(x,Y) = f (0, Y) : 0*f (1, Y) + 0’*f(0,Y) = f(0, Y)

  9. Back to our problem… • X XY’ X’Y (X + Y) X = ? • X (XY’) (X’Y) (X + Y) X = f (X, Y) If X = 1, f (1, Y) = 1 Y’ 0 1 1 = Y If X = 0, f (0, Y) = 0 0 Y Y 0 = 0 Thus, f (X, Y) = XY

  10. 2) NAND, NOR gates NAND, NOR gates are not associative Let a | b = (ab)’ (a | b) | c = a | (b | c)

  11. 3) Block Diagram Transformation a) Reduce # of inputs.  

  12. b. DeMorgan’s Law  (a+b)’ = a’b’  (ab)’ = a’+b’

  13. c. Sum of Products (Using only NAND gates)   Sum of Products (Using only NOR gates)  

  14. d. Product of Sums (NOR gates only)  

  15. Part II. Sequential Networks Memory / Timesteps Clock Flip flops Specification Implementation

More Related